Chapter 4.7, Problem 1CFU

To determine

The reason of why the first step in solving the radical equation given below is to isolate the radical first.

Explanation of Solution

Given information:

The radical equation is $5+\sqrt{x+1}=x$ .

Calculations:

Here the given equation is,

  $5+\sqrt{x+1}=x$

Therefore the first step in solving the equation is to isolate the radical. It is because when we isolate the radical, then the radical comes in one side of the equation. Then by squaring both sides of the equation, we can remove the square root of the radial. After removing the square root we can simplify the equation and then solve it.

Now we solve the equation,

  $\begin{array}{l}5+\sqrt{x+1}=x\\ \Rightarrow \sqrt{x+1}=x-5\\ \Rightarrow {\left(\sqrt{x+1}\right)}^2={\left(x-5\right)}^2\\ \Rightarrow x+1=x^2-10x+25\\ \Rightarrow x^2-11x+24=0\\ \Rightarrow x^2-8x-3x+24=0\\ \Rightarrow x\left(x-8\right)-3\left(x-8\right)=0\\ \Rightarrow (x-3)(x-8)=0\\ Therefore,\\ either,\\ (x-8)=0\\ x=8.\\ or,\\ (x-3)=0\\ x=3.\end{array}$